1.11: Assignment- Argument Essay—First Draft
For this assignment, you will work through the drafting stage of your writing process in an argument essay.
Assignment Instructions
- Review the grading rubric as listed on this page.
-
Choose the position, or side that you agree with, and argue that position using:
- Three separate pieces of evidence, or lines of reasoning, to support your position
- Research to support each line of reasoning (example, testimony, and fact/data, or any combination) that supports your position
- Research about the opposing position’s views
- Presentation of the opposing views, along with lines of reasoning to refute them
- Review the Argument Essay Outline you submitted along with feedback you received from that assignment. Make adjustments to the outline in accordance with the feedback. This may include strengthening lines of reasoning, expanding or improving research sources for supporting evidence, creating a more sympathetic introduction to your position, etc.
-
Develop a complete draft of your essay using your improved Argument Essay Outline. It should meet the following requirements. Papers submitted that do not meet the requirements will be returned to you ungraded.
- A two-part thesis including statement of position; and forecast of your lines of reasoning
- A minimum of three credible outside sources formatted on a source page (MLA, Works Cited)
- MLA formatting
- Minimum of four (4) full pages, not including the Works Cited page .
- Submit your detailed outline as a single file upload.
Requirements
Be sure to:
- Choose a specific issue in which two credible parties have documented clearly opposing positions and clear agendas
-
Choose the position you agree with and argue that position using the following:
- Three separate lines of reasoning;
- Each line of reasoning will support your position with research (example, testimony, and fact/data, or any combination) that supports your position; and
- Research about the opposing position’s views
- Present the opposing views and refute them.
-
Include a two-part thesis with:
- Statement of position; and
- Forecast of your lines of reasoning
- Develop an enticing title that implies your position.
- Use a sympathetic appeal and/or cited research in the introduction to establish the issue.
- Avoid addressing the assignment directly. (Don’t write “I am going to argue about…” Instead, introduce the issue in a more compelling way that makes the reader care.)
- Your voice should be professional and scholarly.
- Package your source material with appropriate signaling and commentary.
Rubric
Grading Rubric: Argument Essay—Draft |
|||
| Criteria | Rating: Meets Expectation | Approaching Expectation | Point Total: 50 |
| Ideas | The paper demonstrates outstanding or above average idea development, with a thesis that articulates a clear stance about a controversial topic. | The writer sufficiently defines the topic, even though development is still basic or general. | __/15 pts |
| Content | The paper demonstrates outstanding or above average evidence of supporting the main point with at least three lives of reasoning. Paragraphs are well-developed and clear, demonstrating research about the topic. Research may include examples, testimony, and fact/data that supports your position. | The writer demonstrates sufficient support of the main point, but could use more supporting details. | __/15 pts |
| Organization | The organization is clear and showcases the central theme. The presentation of information is compelling. Opposing viewpoints are addressed in a logical way, and refuted. | The organizational structure is strong enough to move the reader through the text without too much confusion. | __/15 pts |
| Word Choice, Sentence Fluency, Conventions | The writer demonstrates an outstanding word choice selection, flow and cadence, with well-built sentences and strong grasp of standard writing conventions. | The writer demonstrates sufficient selection of words. The text tends to be more mechanical and contains some errors of standard writing conventions. | __/5 pts |
Contributors and Attributions
CC licensed content, Original
- Provided by : Lumen Learning. Located at : http://lumenlearning.com/ . License : CC BY: Attribution
CC licensed content, Shared previously
- Authored by : Daryl Smith O' Hare and Susan C. Hines. Provided by : Chadron State College. Project : Kaleidoscope Open Course Initiative. License : CC BY: Attribution
- Authored by : Paul Powell. Provided by : Central Community College. Project : Kaleidoscope Open Course Initiative. License : CC BY: Attribution


11.10: Testing Non-normal Data with Wilcoxon Tests
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Okay, suppose your data turn out to be pretty substantially non-normal, but you still want to run something like a t-test? This situation occurs a lot in real life: for the AFL winning margins data, for instance, the Shapiro-Wilk test made it very clear that the normality assumption is violated. This is the situation where you want to use Wilcoxon tests.
Like the t-test, the Wilcoxon test comes in two forms, one-sample and two-sample, and they’re used in more or less the exact same situations as the corresponding t-tests. Unlike the t-test, the Wilcoxon test doesn’t assume normality, which is nice. In fact, they don’t make any assumptions about what kind of distribution is involved: in statistical jargon, this makes them nonparametric tests . While avoiding the normality assumption is nice, there’s a drawback: the Wilcoxon test is usually less powerful than the t-test (i.e., higher Type II error rate). I won’t discuss the Wilcoxon tests in as much detail as the t-tests, but I’ll give you a brief overview.
sample Wilcoxon test
I’ll start by describing the
two sample Wilcoxon test
(also known as the Mann-Whitney test), since it’s actually simpler than the one sample version. Suppose we’re looking at the scores of 10 people on some test. Since my imagination has now failed me completely, let’s pretend it’s a “test of awesomeness”, and there are two groups of people, “A” and “B”. I’m curious to know which group is more awesome. The data are included in the file
awesome.Rdata
, and like many of the data sets I’ve been using, it contains only a single data frame, in this case called
awesome
. Here’s the data:
load("./rbook-master/data/awesome.Rdata")
print( awesome )
## scores group
## 1 6.4 A
## 2 10.7 A
## 3 11.9 A
## 4 7.3 A
## 5 10.0 A
## 6 14.5 B
## 7 10.4 B
## 8 12.9 B
## 9 11.7 B
## 10 13.0 B
As long as there are no ties (i.e., people with the exact same awesomeness score), then the test that we want to do is surprisingly simple. All we have to do is construct a table that compares every observation in group A against every observation in group B. Whenever the group A datum is larger, we place a check mark in the table:
We then count up the number of checkmarks. This is our test statistic, W. 200 The actual sampling distribution for W is somewhat complicated, and I’ll skip the details. For our purposes, it’s sufficient to note that the interpretation of W is qualitatively the same as the interpretation of t or z. That is, if we want a two-sided test, then we reject the null hypothesis when W is very large or very small; but if we have a directional (i.e., one-sided) hypothesis, then we only use one or the other.
The structure of the
wilcox.test()
function should feel very familiar to you by now. When you have your data organised in terms of an outcome variable and a grouping variable, then you use the
formula
and
data
arguments, so your command looks like this:
wilcox.test( formula = scores ~ group, data = awesome)
##
## Wilcoxon rank sum test
##
## data: scores by group
## W = 3, p-value = 0.05556
## alternative hypothesis: true location shift is not equal to 0
Just like we saw with the
t.test()
function, there is an
alternative
argument that you can use to switch between two-sided tests and one-sided tests, plus a few other arguments that we don’t need to worry too much about at an introductory level. Similarly, the
wilcox.test()
function allows you to use the
x
and
y
arguments when you have your data stored separately for each group. For instance, suppose we use the data from the
awesome2.Rdata
file:
load( "./rbook-master/data/awesome2.Rdata" )
score.A
## [1] 6.4 10.7 11.9 7.3 10.0
score.B
## [1] 14.5 10.4 12.9 11.7 13.0
When your data are organised like this, then you would use a command like this:
wilcox.test( x = score.A, y = score.B )
##
## Wilcoxon rank sum test
##
## data: score.A and score.B
## W = 3, p-value = 0.05556
## alternative hypothesis: true location shift is not equal to 0
The output that R produces is pretty much the same as last time.
sample Wilcoxon test
What about the one sample Wilcoxon test (or equivalently, the paired samples Wilcoxon test)? Suppose I’m interested in finding out whether taking a statistics class has any effect on the happiness of students. Here’s my data:
load( "./rbook-master/data/happy.Rdata" )
print( happiness )
## before after change
## 1 30 6 -24
## 2 43 29 -14
## 3 21 11 -10
## 4 24 31 7
## 5 23 17 -6
## 6 40 2 -38
## 7 29 31 2
## 8 56 21 -35
## 9 38 8 -30
## 10 16 21 5
What I’ve measured here is the happiness of each student
before
taking the class and
after
taking the class; the
change
score is the difference between the two. Just like we saw with the t-test, there’s no fundamental difference between doing a paired-samples test using
before
and
after
, versus doing a one-sample test using the
change
scores. As before, the simplest way to think about the test is to construct a tabulation. The way to do it this time is to take those change scores that are positive valued, and tabulate them against all the complete sample. What you end up with is a table that looks like this:
Counting up the tick marks this time, we get a test statistic of V=7. As before, if our test is two sided, then we reject the null hypothesis when V is very large or very small. As far of running it in R goes, it’s pretty much what you’d expect. For the one-sample version, the command you would use is
wilcox.test( x = happiness$change,
mu = 0
)
##
## Wilcoxon signed rank test
##
## data: happiness$change
## V = 7, p-value = 0.03711
## alternative hypothesis: true location is not equal to 0
As this shows, we have a significant effect. Evidently, taking a statistics class does have an effect on your happiness. Switching to a paired samples version of the test won’t give us different answers, of course; but here’s the command to do it:
wilcox.test( x = happiness$after,
y = happiness$before,
paired = TRUE
)
##
## Wilcoxon signed rank test
##
## data: happiness$after and happiness$before
## V = 7, p-value = 0.03711
## alternative hypothesis: true location shift is not equal to 0


2.2: Basic Communication Concepts
Effective communication is one of the Standards of Professional Performance established by the American Nurses Association. The standard states, “The registered nurse communicates effectively in all areas of practice.” [1] There are several concepts related to effective communication such as demonstrating appropriate verbal and nonverbal communication, using assertive communication, being aware of personal space, and overcoming common barriers to effective communication.
Types of Communication
Verbal Communication
Effective communication requires each interaction to include a sender of the message, a clear and concise message, and a receiver who can decode and interpret that message. The receiver also provides a feedback message back to the sender in response to the received message. See Figure 2.1 [2] for an image of effective communication between a sender and receiver.
Nurses assist patients and their family members to understand health care needs and treatments by using verbal, nonverbal, and written communication. Verbal communication is more than just talking. Effective verbal communication is defined as an exchange of information using words understood by the receiver in a way that conveys professional caring and respect. [3] Nurses who speak using extensive medical jargon or slang may create an unintended barrier to their own verbal communication processes. When communicating with others, it is important for the nurse to assess the receiver’s preferred method of communication and individual receiver characteristics that might influence communication, and subsequently adapt communication to meet the receiver’s needs. For example, the nurse may adapt postsurgical verbal instruction for a pediatric versus an adult patient. Although the information requirements regarding signs of infection, pain management, etc., might be similar, the way in which information is provided may be quite different based on developmental level. Regardless of the individual adaptations that are made, the nurse must be sure to always verify patient understanding.
Nonverbal Communication
In addition to communicating verbally, the nurse must also be aware of messages sent by nonverbal communication . Nonverbal communication can have a tremendous impact on the communication experience and may be much more powerful than the verbal message itself. You may have previously learned that 80% of communication is nonverbal communication (see Figure 2.2 [4] ). The importance of nonverbal communication during communication has also been described in percentages of 55, 38, and 7, meaning 55% of communication is body language, 38% is tone of voice, and 7% is the actual words spoken. [5]
Nonverbal communication includes body language and facial expressions, tone of voice, and pace of the conversation. For example, compare the nonverbal communication messages in Figures 2.3 [6] and 2.4. [7] What nonverbal cues do you notice about both toddlers?
Nurses should be attentive to their nonverbal communication cues and the messages they provide to patients and their families. Nurses should be purposeful in their use of nonverbal communication that conveys a feeling of caring. [8] What nonverbal cues do you notice about the nurse in Figure 2.5 [9] that provide a perception of professional caring?
Nurses use nonverbal communication such as directly facing patients at eye level, leaning slightly forward, and making eye contact to communicate they care about what the person is telling them and they have their full attention. [10]
It is common for health care team members in an acute care setting to enter a patient’s room and begin interacting with a patient who is seated or lying in bed. However, it is important to remember that initial or sensitive communication exchanges are best received by the patient if the nurse and patient are at eye level. Bringing a chair to the patient’s bedside can help to facilitate engagement in the communication exchange. SOLER is common mnemonic used to facilitate nonverbal communication ( s it with o pen posture and l ean in with good e ye contact in a r elaxed manner).
Communication Styles
In addition to verbal and nonverbal communication, people communicate with others using three styles. A passive communicator puts the rights of others before their own. Passive communicators tend to be apologetic or sound tentative when they speak and often do not speak up if they feel as if they are being wronged. Aggressive communicators, on the other hand, come across as advocating for their own rights despite possibly violating the rights of others. They tend to communicate in a way that tells others their feelings don’t matter. However, assertive communicators respect the rights of others while also standing up for their own ideas and rights when communicating. An assertive person is direct, but not insulting or offensive. [11] Assertive communication refers to a way of conveying information that describes the facts and the sender’s feelings without disrespecting the receiver’s feelings. Using “I” messages such as, “I feel…,” “I understand…,” or “Help me to understand…” are strategies for assertive communication. This method of communicating is different from aggressive communication that uses “you” messages and can feel as if the sender is verbally attacking the receiver rather than dealing with the issue at hand. For example, instead of saying to a coworker, “Why is it always so messy in your patients’ rooms? I dread following you on the next shift!,” an assertive communicator would use “I” messages to say, “I feel frustrated spending the first part of my shift decluttering our patients’ rooms. Help me understand why it is a challenge to keep things organized during your shift?”
Using assertive communication is an effective way to solve problems with patients, coworkers, and health care team members.
View this humorous video demonstrating assertive communication techniques being used by the actors on a TV show:
Everybody Loves Raymond Uses Active Listening – from Parent Effectiveness Training
.
Personal Space
While being aware of verbal and nonverbal messages and communicating assertively, it is also important to be aware of others’ personal space. Proxemics is the study of personal space and provides guidelines for professional communication. The public zone is over 10 feet of distance between people and generally avoids physical contact. The social zone is four to 10 feet of distance between people. It is used during social interactions and business settings. The personal zone is 18 inches to four feet of space and is generally reserved for friends and family. Less than 18 inches is reserved for close relationships but may be invaded when in crowds or playing sports. [12] Nurses usually communicate within the social zone to maintain professional boundaries. However, when assessing patients and performing procedures, nurses often move into a patient’s personal zone. Nurses must be aware of patients’ feelings of psychological discomfort that can occur when invading this zone. Additionally, cultural considerations may impact the appropriateness of personal space when providing patient care. See Figure 2.6 for example of personal space zones. [13]
Overcoming Common Barriers to Communication
It is important for you to reflect on personal factors that influence your ability to communicate effectively. There are many factors that can cause the message you are trying to communicate to become distorted and not perceived by the receiver in the way you intended. It is important to seek feedback that your message is clearly understood. Nurses must be aware of these potential barriers and try to reduce their impact by continually seeking feedback and checking understanding. [14]
Common barriers to communication in health care and strategies to overcome them are described in the following box. [15]
- Jargon: Avoid using medical terminology, complicated, or unfamiliar words. When communicating with patients, explain information in plain language that is easy to understand by those without a medical or nursing background.
- Lack of attention: Nurses are typically very busy with several tasks to complete for multiple patients. It is easy to become focused on the tasks instead of the patient. When entering a patient’s room, it is helpful to pause, take a deep breath, and mindfully focus on the patient in front of you to give them your full attention. Patients should feel as if they are the center of your attention when you are with them, no matter how many other things you have going on.
- Noise and other distractions: Health care environments can be very noisy with people talking in the room or hallway, the TV blaring, alarms beeping, and pages occurring overhead. Create a calm, quiet environment when communicating with patients by closing doors to the hallway, reducing the volume of the TV, or moving to a quieter area, if possible.
- Light: A room that is too dark or too light can create communication barriers. Ensure the lighting is appropriate according to the patient’s preference.
- Hearing and speech problems: If your patient has hearing or speech problems, implement strategies to enhance communication. See the “Adapting Your Communication” section below for strategies to address hearing and speech problems.
- Language differences: If English is not your patient’s primary language, it is important to seek a medical interpreter and to also provide written handouts in the patient’s preferred language when possible. Most agencies have access to an interpreter service available by phone if they are not available on-site.
- Differences in cultural beliefs: The norms of social interaction vary greatly in different cultures, as well as the ways that emotions are expressed. For example, the concept of personal space varies among cultures, and some patients are stoic about pain whereas others are more verbally expressive. Read more about caring for diverse patients in the “ Diversity ” chapter.
- Psychological barriers: Psychological states of the sender and the receiver affect how the message is sent, received, and perceived. For example, if nurses are feeling stressed and overwhelmed with required tasks, the nonverbal communication associated with their messages such as lack of eye contact, a hurried pace, or a short tone can affect how the patient perceives the message. If a patient is feeling stressed, they may not be able to “hear” the message or they may perceive it differently than it was intended. It is important to be aware of signs of the stress response in ourselves and our patients and implement appropriate strategies to manage the stress response. See the box below for more information about strategies to manage the stress response.
- Physiological barriers: It is important to be aware of patients’ potential physiological barriers when communicating. For example, if a patient is in pain, they are less likely to hear and remember what was said, so pain relief should be provided as needed before providing patient education. However, it is also important to remember that sedatives and certain types of pain medications often impair the patient’s ability to receive and perceive messages so health care documents cannot be signed by a patient after receiving these types of medications.
- Physical barriers for nonverbal communication: Providing information via e-mail or text is often less effective than face-to-face communication. The inability to view the nonverbal communication associated with a message such as tone of voice, facial expressions, and general body language often causes misinterpretation of the message by the receiver. When possible, it is best to deliver important information to others using face-to-face communication so that nonverbal communication is included with the message.
- Differences in perception and viewpoints: Everyone has their own beliefs and perspectives and wants to feel “heard.” When patients feel their beliefs or perspectives are not valued, they often become disengaged from the conversation or the plan of care. Nurses should provide health care information in a nonjudgmental manner, even if the patient’s perspectives, viewpoints, and beliefs are different from their own.
The stress response is a common psychological barrier to effective communication. It can affect the message sent by the sender or how it is received by the receiver. The stress response is a common reaction to life events, such as a nurse feeling stressed by being overwhelmed with tasks to complete for multiple patients, or a patient feeling stressed when admitted to a hospital or receiving a new diagnosis. Symptoms of the stress response include irritability, sweaty palms, a racing heart, difficulty concentrating, and impaired sleep. It is important to recognize symptoms of the stress response in ourselves and our patients and use strategies to manage the stress response when communicating. Strategies to manage the stress response are as follows:
- Use relaxation breathing . Become aware of your breathing.  Repeat this process at least three times in succession and then as often as needed throughout the day.
- Make healthy diet choices. Avoid caffeine, nicotine, and junk food because these items can increase feelings of anxiety or being on edge.
- Make time for exercise. Exercise stimulates the release of natural endorphins that reduce the body’s stress response and also helps to improve sleep.
- Get enough sleep. Set aside at least 30 minutes before going to bed to wind down from the busyness of the day. Avoid using electronic devices like cell phones before bedtime because the backlight can affect sleep.
-
Use
progressive relaxation
. There are several types of relaxation techniques that focus on reducing muscle tension and using mental imagery to induce calmness. Progressive relaxation generally includes the following steps:
- Start by lying down somewhere comfortable and firm, like a rug or mat on the floor. Get yourself comfortable.
- Relax and try to let your mind go blank. Breathe slowly, deeply, and comfortably, while gradually and consciously relaxing all your muscles, one by one.
- Work around the body one main muscle area at a time, breathing deeply, calmly, and evenly. For each muscle group, clench the muscles tightly and hold for a few seconds, and then relax them completely. Repeat the process, noticing how it feels. Do this for each of your feet, calves, thighs, buttocks, stomach, arms, hands, shoulders, and face.
- American Nurses Association. (2015). Nursing: Scope and standards of practice (3rd ed.). American Nurses Association. ↵
- “Osgood-Schramm-model-of-communication.jpg“ by Jordan Smith at eCampus Ontario is licensed under CC BY 4.0 . Access for free at https://ecampusontario.pressbooks.pub/communicationatwork/chapter/1-3-the-communication-process/ ↵
- This work is a derivative of Human Relations by LibreTexts and is licensed under CC BY-NC-SA 4.0 ↵
- “ Constituents of Communication.png ” by jb11ko, lb13an, ad14xz, jb12xu is licensed under CC BY-SA 4.0 ↵
- Thompson, J. (2011). Is nonverbal communication a numbers game? Psychology Today. https://www.psychologytoday.com/us/blog/beyond-words/201109/is-nonverbal-communication-numbers-game ↵
- “ I’m angry ” by WiLPrZ is licensed under CC BY-NC-ND 2.0 ↵
- “ Happy Toddler ” by Chris Bloom is licensed under CC BY-SA 2.0 ↵
- This work is a derivative of Human Relations by LibreTexts and is licensed under CC BY-NC-SA 4.0 ↵
- " PIXNIO-42752-4542x3003.jpg " by James Gathany, Judy Schmidt, USCDCP is in the Public Domain ↵
- Stickley, T. (2011). From SOLER to SURETY for effective non-verbal communication. Nurse Education in Practice, 11 (6), 395-398. https://doi.org/10.1016/j.nepr.2011.03.021 ↵
- This work is a derivative of Human Relations by LibreTexts and is licensed under CC BY-NC-SA 4.0 ↵
- Psychology Today. (n.d.) Proxemics. https://www.psychologytoday.com/us/basics/proxemics ↵
- “ Personal Space.svg ” by WebHamster is licensed under CC BY-SA 3.0 ↵
- SkillsYouNeed. (n.d.). Barriers to effective communication . https://www.skillsyouneed.com/ips/barriers-communication.html ↵
- SkillsYouNeed. (n.d.). Barriers to effective communication . https://www.skillsyouneed.com/ips/barriers-communication.html ↵
- American Psychological Association. (2019). Healthy Ways to Handle Life's Stressors. https://www.apa.org/topics/stress/tips ↵


16.2: Cohesion- What do People Mean When They Say My Writing Doesn’t “Flow”
Before You Read
Write 750 to 1,000 words in response to this prompt, keeping in mind that you’re going to be working with this piece of writing extensively in the coming days—not only in this chapter but in other chapters related to style:
As a public university that receives substantial support from the state legislature, the U has an obligation to serve the people of the state of Utah. However, there can be disagreements on what that service means. On one hand, it can mean admitting as many Utah residents as possible in order to increase the state’s population of college-educated citizens. On the other hand, it can mean increasing admission requirements to fulfill the U’s position as the “flagship” university in the state. Where do you come down?
Cohesion, or “Flow”—An Overview
Figure \(\PageIndex{1}\)
Many (student) writers have turned in papers only to have their readers (more often than not, their teachers) hand the papers back with comments that the writing doesn’t “flow.”
Unfortunately, teachers may not always be explicit about what they mean—just that it doesn’t “read” or “sound” right or that the ideas don’t progress from one to another. This chapter is about what “flow” actually means and how to make sure your writing does it.
By “flow,” most readers mean what grammarians and linguists call cohesion —the property of a text to hold together at the level of sentences and paragraphs. Of course, cohesion is good in any communication medium, and each medium can present challenges for it. If you’re sending text messages back and forth to a friend and the network sends them out of order, the result can be confusing: you might have written “thank GOD” in response to some piece of news, but your friend might not have gotten your message until after she texted “gotta go.” Oops. If you’re speaking to someone on a train or bus and something outside the window catches your attention, you might say something about it, and the other person might say, “wait—what?”
But in both those cases, you can quickly and easily clear up the confusion. Speech and texting are more or less synchronous media: that is, they involve people communicating at the same time and often in the same (virtual) space. Writing, however—in the traditional sense, anyway—is different, because it’s asynchronous . It also requires an important trade-off. Writing has worked well for a long time as a communication technology because it’s relatively easy to distribute. Someone using writing to communicate doesn’t have to move from place to place: she may simply write something down and send it. However, to use a metaphor from very current communication technologies, writing has low bandwidth compared to other media. If someone is speaking to you, you can infer meaning from words themselves but also from vocal inflections, facial expressions, hand gestures, posture, and even from how close the other person is to you. You can’t do that when you write and read. So, writers and readers can send and receive on the cheap, but they carry a burden of making their words work extraordinarily hard.
This idea has a very clear implication for your own written arguments—an important enough implication that we’d say it pretty loudly if you were standing right in front of us. But, since this is writing, we’ll use boldface: just because an argument you’re making is clear in your own head, that doesn’t mean it’s automatically clear to people who are reading the written version of your argument . That’s one of the reasons it’s a good idea to circulate the writing you do to others before you turn it in for a grade or circulate it in high-stakes situations.
Fortunately, the to-do list for “flow” is relatively short. Throughout Englishlanguage writing, it turns out that there is a small number of strategies for achieving cohesion. These strategies help writers follow a key principle for communicating with readers as effectively as possible on the assumption that they’re not looking over their readers’ shoulders pointing out what they really need to know. That principle is called the given-new contract . This contract implies that you as a writer will start your projected readers with something relatively familiar and then lead them to less familiar material. It’s an idea that is simple to state, but it’s powerful, and it works at different levels of a document. At the level of overall document design, consistent visual items on each page (page number location, headings, “white” space, fonts) help create a familiar visual field that works like a container for whatever new information is coming next. As you read earlier in this book, a lot of a writer’s job in an introduction, after all, is orienting readers so that they’re at least familiar with the broad topic before the writer gets specific—with an argument, for example. But the contract helps sentence-level cohesion, as well. It’s very helpful to readers if you create a cycle in which you try to put “given” information at the start of sentences and shift “new” information to the ends, and then recycle the “new” information as “given” information in sentences that come up. The principle of end emphasis helps here: readers tend to latch more onto how sentences end than onto how they begin. Skilled writers know this is often the case, so they’ll reserve end-of-sentence slots for new or challenging information, since they know they often have a little more of their readers’ attention at those spots anyway.
The given-new contract and the concept of end emphasis are a little tough to explain in abstract terms, so here’s an example followed by some analysis. We’ve numbered the sentences to help make the analysis clear.
1 This textbook is freely circulable under the terms of a Creative Commons (“CC”) license. 2 CC is a nonprofit organization that helps content creators, such as textbook authors, share their products in more diverse ways than traditional copyright allows. 3 While typical copyright restricts others from using an author’s work unless they have the author’s express consent, CC allows authors to pick and choose which restrictions to apply to their work by using one of several free licenses. 4 For example, this book is available via an “Attribution-NonCommercialShareAlike” agreement: adopters of the textbook may use it free of charge and may even modify it without permission, but they must agree not to try to sell it or share it with others under different licensing terms.
Each sentence in this passage shows our attempt to honor the given-new contract. Here’s how:
- The first sentence introduces the term “Creative Commons” near its end. We’re assuming that you may not know (much) about CC, so we’re trying to exploit end emphasis to introduce it here very early in the paragraph.
- This sentence immediately recycles CC and defines the term more fully. The sentence ends with the important (and “new”) idea that CC allows for a wider range of options than copyright.
- Now, the passage explains in a little more detail the point it just made about copyright restrictions and goes on to clarify the contrast with CC, ending with the “new” information that CC allows authors to choose from several licenses.
- Not surprisingly, the next sentence shows what the previous sentence introduced by giving an example of a relevant CC license.
In addition to using the principle of end emphasis, writers who honor the given-new contract frequently use several other strategies.
Stock transition words and phrases
Many writers first learn to make their writing flow by using explicit, specialpurpose transitional devices. You may hear these devices called “signposts,” because they work much like highway and street signs. When steel boxes weighing 2 tons and more are rolling around at high speeds, it’s important that their operators are repeatedly and clearly told exactly where and when to go with as little ambiguity as possible.
Here’s a list of stock, generic, all-purpose transition words and phrases, organized by their basic functions. Keep in mind that there are differences among these that can make a difference and that determining what those differences is is beyond the scope of this book. It’s a matter of experience.
Figure \(\PageIndex{2}\)
To add or show sequence : again, also, and, and then, besides, equally important, finally, first, further, furthermore, in addition, in the first place, last, moreover, next, second, still, too
To compare : also, in the same way, likewise, similarly
To contrast : although, and yet, but, but at the same time, despite, even so, even though, for all that, however, in contrast, in spite of, nevertheless, notwithstanding, on the contrary, on the other hand, regardless, still, though, yet
To give examples or intensify : after all, an illustration of, even, for example, for instance, indeed, in fact, it is true, of course, specifically, that is, to illustrate, truly
To indicate place : above, adjacent to, below, elsewhere, farther on, here, near, nearby, on the other side, opposite to, there, to the east, to the left
To indicate time : after a while, afterward, as long as, as soon as, at last, at length, at that time, before, earlier, formerly, immediately, in the meantime, in the past, lately, later, meanwhile, now, presently, shortly, simultaneously, since, so far, soon, subsequently, then, thereafter, until, until now, when
To repeat, summarize, or conclude : all in all, altogether, in brief, in conclusion, in other words, in particular, in short, in simpler terms, in summary, on the whole, that is, to put it differently, to summarize
To show cause and effect : accordingly, as a result, because, consequently, for this purpose, hence, otherwise, since, then, therefore, thereupon, thus, to this end, with this object in mind
As we just told you, avoiding ambiguity in academic and professional writing is important. But it’s not as important as avoiding it on highways, in factories, or around high-voltage equipment or explosives. In those contexts, lots of signposts with lots of redundancy are vital. In many writing situations, you can expect your readers to pick up other useful clues for cohesion, so it’s somewhat less important to use a lot of these “stock” or generic transition words. In fact, if you overuse them (for instance, in an essay in which your first paragraph starts with “first,” your second paragraph starts with “second,” and on and on), it can get annoying.
Pronouns
If you’re old enough vaguely to remember the Schoolhouse Rock series, you might remember the episode about pronouns (“he,” “she,” “her,” “him,” “you,” “we,” “they,” “it,” “one,” “this,” “that,” and some others) and how they can stand for nouns, even if the nouns have long names. The idea is that pronouns make speaking and writing more efficient. But you may not have learned that pronouns are at least as powerful as cohesive devices. Since pronouns work by referring back to nouns that have previously been mentioned, they can help writers carry the ideas their nouns represent across sentences and paragraphs.
You may have been told to limit your use of pronouns or even avoid them altogether. This is bad advice, but it’s understandable: pronouns work very well when they clearly refer to their antecedents, but they can create significant comprehension problems, misdirection, and vagueness when they don’t.
Repetition
Contrary to a lot of advice novice writers get, repetition is effective. For example, as you’ll learn later in this book (or now if you want to read ahead, of course), many rhetorical strategies that are thousands of years old and that exist in several languages use repetition. It’s a time-honored way to signal importance, create a sense of rhythm, and help audiences remember key ideas. But repetition gets a bad reputation because it can become redundant. (Yes, that sentence used repetition to get its point across. It’s no accident that it had a lot of “r”s.)
Figure \(\PageIndex{3}\)
Repetition can involve individual words, phrases, or grammatical structures. When you repeat similar structural elements but not necessarily the words themselves, you are using parallelism , a special variety of repetition that not only helps cohesion but also helps you to communicate that similarly important ideas should be read together. When sentences are written using non-parallel parts, it’s certainly possible for readers to understand them, but it creates work for the reader that usually isn’t necessary. Compare these sentences:
Student writers should learn to start projects early, how to ask for advice from teachers and peers, and when to focus on correcting grammar.
Student writers should learn to start projects early, to ask for advice from teachers and peers, and to figure out when to focus on correcting their grammar.
See the difference? The first sentence is comprehensible: the commas, for example, let you know that you’re reading a list. But the extra adverbs (“how” and “when”) get in the way of the sentence’s clarity. And that problem, in turn, means that it’s hard to see clearly how each item in the list relates to the others. In the revised sentence, though, it’s a lot clearer that each of the three items is something student writers should “learn to” do. That relationship is made clear by the repeating grammatical pattern:
Student writers should learn
- to start projects early
- to ask for advice
- to figure out when to focus on grammar
Example
Here’s an example of some writing that uses a variety of cohesion strategies. We know it well because one of us wrote it. It’s a short essay, written for a broad academic audience in a U publication, about the current state of the English language. To clarify the analysis that follows, we’ve underlined a few of the transition devices.
Teaching (and Learning) Englishes
Jay Jordan
University Writing Program
I teach English-language writing, and I’m a native speaker of the English language. Being a native speaker might seem to be an excellent basic qualification for my job: at the very least, it should necessarily make me the model of English usage. However, it actually makes me very unusual.
According to The British Council, approximately 1.5 billion people around the world use English. Roughly 375 million of them are like me: they have learned English since birth, and most of them live in countries like the US, Canada, Great Britain, Australia, and New Zealand that are traditional English-language centers.
That still leaves over a billion English users. 375 million of those people live in countries that were British colonies until the middle of the last century, such as Ghana, India, Kenya, and Nigeria.
But the largest number of English speakers—50% of the global total—are in countries that were not British colonies and that don’t have much of a history with English. Count China, Indonesia, Japan, Malaysia, and South Korea among them. So, most English speakers aren’t where we might expect them to be. In addition, they’re not using English in ways we might expect, either, which helps explain why I’m referring to them as “users” and not “writers” or “speakers.” Most people who use English around the world do so in specific circumstances in order to get very specific things done. Many Indians, for instance, might use English in publications and to transact business over the phone, Hindi in a government office, Gujarati at the store, and maybe one of several other languages at home.
What does this mean for my teaching and research? People and information move around globally more so now than ever, and that movement makes diverse uses of English feed back into the US. As students at the U (and the U is not alone) become more culturally and linguistically diverse, I often have as much to learn from them as I have to teach them.
This short example uses each of the cohesion strategies described above:
- Overall, the example attempts to honor the given-new contract. It starts on familiar territory—or at least, with an attempt to orient the reader very quickly to the writer’s personal approach. And it also makes a statement about the writer that the reader likely intuitively agrees with: namely, he’s a native speaker of English, which makes him well qualified to be an English teacher. But the first paragraph ends with a surprising claim: being a native English speaker means being unusual. Here, then, the writer starts with what’s comfortable but then uses end emphasis to reinforce the “new” information at the end.
- The writer does use several stock transitions: in fact, one of them —“however”—helps introduce the surprising sentence at the end of the introduction by clearly signposting something different or unexpected. And, as another example, the fourth paragraph starts with “but,” which signposts another transition to information that contradicts what comes before. (You may have been told never to start sentences with conjunctions like “but” or “and.” It turns out that it’s generally fine to do that. Just be aware of your readers’ preferences.)
- Pronouns appear to be the most common cohesion device in the essay. At the start of the third paragraph, for example, “that” stands in for the statistic in the previous paragraph, which would be hard to write out all over again. But “that” also carries forward the sense of the statistic into the next paragraph. And “those people” carries the statistic forward to the next sentence. (Really, “those” is actually an adjective that modifies “people,” but it’s enough like a pronoun that we’re handling it like one here.)
- Repetition is also common in this essay. Words are repeated—or at least, put very close to other words that are very similar in meaning. “English” and “British colonies” clearly help tie together the third and fourth paragraphs. And sentences show parallelism. See, for instance, paragraph four: “So, most English speakers aren’t where we would expect them to be. In addition, they’re not using English in ways we might expect, either.”
To Do
- Identify at least three other specific cohesion devices used in the example essay. Be prepared to say what kind of device it is and what effect it has on your reading. Also be prepared to suggest what would happen if it weren’t there.
- Re-read the 750-1,000 words you wrote before you read this chapter, paying particular attention to cohesion. Now, revise it to improve its flow.


5.7: Functions
- Use function notation.
- Determine if a relation is a function with different representations.
- Apply the vertical line test.
- Determine the domain and range of a function.
In this section, we will learn about relations and functions. As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number, and our birthday are matched to our name. There is a relationship between our name and each of those items. When your teacher gets their class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.
(Student name, Student ID #)
The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation. In general terms, a relation is any set of ordered pairs, ( ). All the \(x\)-values in the ordered pairs together make up the domain . All the -values in the ordered pairs together make up the range.
There are many situations similar to the student's name and student ID # where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation. A special type of relation, called a function , occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each \(x\)-value is matched with only one -value.
Let us look at the relation between your friends and their birthdays in Figure
. Every friend has a birthday, but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24 as their birthday and that June and Liz share August 2. Since each person has exactly one birthday, the relation is a function.Use Function Notation
It is very convenient to name a function; most often functions are named \(f, g, h, F, G\), or \(H\). In any function, for each \(x\)-value from the domain, we get a corresponding \(y\)-value in the range. In the function \(f\), we write this range value \(y\) as \(f(x)\). This notation \(f(x)\) is called function notation and is read "f of \(x\) " or "the value of f at \(x\)." In this case the parentheses do not indicate multiplication.
We call \(x\) the independent variable as it can be any value in the domain. We call \(y\) the dependent variable as its value depends on \(x\). Much like when you first encountered the variable \(x\), function notation may be rather unsettling. But the more you use the notation, the more familiar you become with the notation, and the more comfortable you will be with it.
Let's review the equation \(y=4 x-5\). To find the value of \(y\) when \(x=2\), we know to substitute \(x=2\) into the equation and then simplify.
| \(y=4 x-5\) | |
| Let \(x=2\). | \(\begin{array}{l}y=4 \cdot 2-5 \\ y=3\end{array}\) |
The value of the function at \(x=2\) is 3 . We do the same thing using function notation, the equation \(y=4 x-5\) can be written as \(f(x)=4 x-5\). To find the value when \(x=2\), we write:
| \(f(x)=4 x-5\) | |
| Let \(x=2\). | \(\begin{array}{l}f(2)=4 \cdot 2-5 \\ f(2)=3\end{array}\) |
The value of the function at is 3. This process of finding the value of for a given value of \(x\) is called evaluating the function .
For the function \(f(x)=2 x^2+3 x-1\), evaluate the function.
- \(f(3)\)
- \(f(-2)\)
- \(f(a)\)
- Answer
-
1. To evaluate \(f(3)\), substitute 3 , for \(x\).
Simplify.
\[\begin{array}{l}
f(x)=2 x^2+3 x-1 \\
f(\mathbf{3})=2(\mathbf{3})^2+3 \cdot \mathbf{3}-1 \\
f(3)=2 \cdot 9+3 \cdot 3-1 \\
f(3)=18+9-1 \\
f(3)=26
\end{array} \nonumber \]2. To evaluate \(f(-2)\), substitute -2 for \(x\).
Simplify.
\[\begin{aligned}
f(x) & =2 x^2+3 x-1 \\
f(-2) & =2(-2)^2+3(-2)-1 \\
f(-2) & =2 \cdot 4+(-6)-1 \\
f(-2) & =8+(-6)-1 \\
f(-2) & =1
\end{aligned} \nonumber \]3. To evaluate \(f(a)\), substitute \(a\) for \(x\).
Simplify.
\[\begin{array}{l}
f(x)=2 x^2+3 x-1 \\
f(\boldsymbol{a})=2(\boldsymbol{a})^2+3 \cdot \boldsymbol{a}-1 \\
f(a)=2 a^2+3 a-1
\end{array} \nonumber \]
For the function \(f\left( x \right) = 3{x^2} - 2x + 1\), evaluate the function.
- \(f\left( 3 \right)\)
- \(f\left( { - 1} \right)\)
- \(f\left( t \right)\)
The number of unread emails in Sylvia's inbox is 75 . This number grows by 10 unread emails a day. The function \(N(t)=75+10 t\) represents the relation between the number of emails, \(N\), and the time, \(t\), measured in days. Find \(N(5)\). Explain what this result means.
- Answer
-
Find \(N(5)\). Explain what this result means.
Substitute in \(t=5\).
Simplify.\[\begin{aligned}
N(\mathbf{5}) & =75+10 \cdot \mathbf{5} \\
N(t) & =75+10 t \\
N(5) & =75+50 \\
N(5) & =125
\end{aligned} \nonumber \]If 5 is the number of days, \(N(5)\) is the number of unread emails after 5 days. After 5 days, there are 125 unread emails in Sylvia's inbox.
The number of unread emails in Bryan’s account is 100. This number grows by 15 unread emails a day. The function \(N\left( t \right) = 100 + 15t\) represents the relation between the number of emails, \(N\), and the time, \(t\), measured in days. Find \(N\)(7). Explain what the result means.
Determining If a Relation Is a Function with Different Representations
We can determine whether a relation is a function by identifying the input and the output values. If each input value leads to only one output value, classify the relation as a function. If any input value leads to two or more outputs, do not classify the relation as a function.
We will review three different representations of relations and determine if they are functions: ordered pairs, mapping, and equations.
Use the set of ordered pairs to determine whether the relation is a function.
- \(\{(-3,27),(-2,8),(-1,1),(0,0),(1,1),(2,8),(3,27)\}\)
- \(\{(9,-3),(4,-2),(1,-1),(0,0),(1,1),(4,2),(9,3)\}\)
- Answer
-
- Each \(x\)-value is matched with only one -value. This relation is a function.
- The \(x\)-value 9 is matched with two \(y\)-values, both 3 and -3. This relation is not a function.
Use the set of ordered pairs to determine whether the relation is a function.
- \(\left\{ {\left( {-3,-6} \right),\left( {-2,-4} \right),\left( {-1,-2} \right),\left( {0,0} \right),\left( {1,2} \right),\left( {2,4} \right),\left( {3,6} \right)} \right\}\)
- \(\{(8,-4),(4,-2),(2,-1),(0,0),(2,1),(4,2),(8,4)\}\)
A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range. Consider the example of the relation between your friends and their birthdays used in Figure 5.60. In this particular example, the domain is the set of people’s names, and the range is the set of their birthdays. This mapping was a function because everybody’s name maps to exactly one birthday.
Use the mapping in Figure
to determine whether the relation is a function.- Answer
-
Both Lydia and Marty have two phone numbers. Each \(x\)-value is not matched with only one -value. This relation is not a function.
Use the mapping in the given figure to determine whether the relation is a function.
In algebra functions will usually be represented by an equation. It is easiest to see if the equation is a function when it is solved for . If each value of \(x\) results in only one value of , then the equation defines a function.
Determine whether each equation is a function. Assume \(x\) is the independent variable.
1. \(2 x+y=7\)
2. \(y=x^2+1\)
3. \(x+y^2=3\)
- Answer
-
1. \(2 x+y=7\)
For each value of \(x\), we multiply it by -2 and then add 7 to get the \(y\)-value.
For example, if \(x=3\) :\[\begin{array}{l}
y=-2 x+7 \\
y=-2 \cdot \mathbf{3}+7 \\
y=1
\end{array} \nonumber \]We have that when \(x=3\), then \(y=1\). It would work similarly for any value of \(x\). Since each value of \(x\), corresponds to only one value of \(y\) the equation defines a function.
2. \(y=x^2+1\)
For each value of \(x\), we square it and then add 1 to get the \(y\)-value.
For example, if \(x=2\)\[\begin{array}{l}
y=x^2+1 \\
y=\mathbf{2}^2+1 \\
y=5
\end{array} \nonumber \]We have that when \(x=2\), then \(y=5\). It would work similarly for any value of \(x\). Since each value of \(x\) corresponds to only one value of \(y\), the equation defines a function.
3. \(x+y^2=3\)
\[x+y^2=3 \nonumber \]
Isolate the \(y\) term.
\[y^2=-x+3 \nonumber \]
Let us substitute \(x=2\).
\[\begin{array}{l}
y^2=-\mathbf{2}+3 \\
y^2=1
\end{array} \nonumber \]This gives us two values for \(y\).
\[y=1, y=-1 \nonumber \]
We have shown that when \(x=2\), then \(y=1\) and \(y=-1\). It would work similarly for any value of \(x\). Since each value of \(x\) does not corresponds to only one value of \(y\) the equation does not define a function.
Determine whether each equation is a function.
1. \(\backslash(4 x+y=-3 \backslash)\)
2. \(\backslash\left(x+\left\{y^{\wedge} 2\right\}=1 \backslash\right)\)
3. \(\backslash\left(y-\left\{x^{\wedge} 2\right\}=2 \backslash\right)\)
Relations and Functions
Applying the Vertical Line Test
We reviewed how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping, or an equation. We will now cover how to tell if a graph is that of a function.
An ordered pair \((x, y)\) is a solution of a linear equation, if the equation is a true statement when the \(x\)-values and \(y\) -values of the ordered pair are substituted into the equation. The graph of a linear equation is a straight line where every point on the line is a solution of the equation, and every solution of this equation is a point on this line. Figure
A relation is a function if every element of the domain has exactly one value in the range. The relation defined by the equation vertical line test , which states that a set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.
Determine whether the graph (Figure
) is the graph of a function applying the vertical line test.- Answer
-
On the graph (Figure
), only three vertical dashed lines are drawn. However, it can be determined that any vertical dashed line that is drawn will intersect the solid line at exactly one point. It is the graph of a function.
Determine whether the graph is the graph of a function.
Determine whether the graph is the graph of a function (Figure
).Determine whether the graph is the graph of a function.
Determining the Domain and Range of a Function
For the function \( y = f(x)\), \(x\) is the independent variable as it can be any value in the domain, and \(y\) is the dependent variable since its value depends on \(x\). For the function \( y = f(x)\), the values of \(x\) make up the domain and the values of \(y\) make up the range.
For \(\{(1,1),(2,4),(3,9),(4,16),(5,25)\}\):
- Find the domain of the relation.
- Find the range of the relation.
- Answer
-
1. The domain is the set of all \(x\)-values of the relation: \(\{1,2,3,4,5\}\)
2. The range is the set of all \(y\)-values of the relation: \(\{1,4,9,16,25\}\)
For the relation \(\left\{ {\left( {1,1} \right),\left( {2,8} \right),\left( {3,27} \right),\left( {4,64} \right),\left( {5,125} \right)} \right\}\):
- Find the domain of the relation.
- Find the range of the relation.
Use Figure \(\PageIndex{11}\) to:
- List the ordered pairs of the relation.
- Find the domain of the relation.
- Find the range of the relation.
- Answer
-
1. The ordered pairs of the relation are: \(\{(1,5),(-3,-1),(4,-2),(0,3),(2,-2),(-3,4)\}\).
2. The domain is the set of all \(x\)-values of the relation: \(\{-3,0,1,2,4\}\). Notice that while -3 repeats, it is only listed once.
3. The range is the set of all \(y\)-values of the relation: \(\{-2,-1,3,4,5\}\). Notice that while -2 repeats, it is only listed once.
Use the given figure to:
- List the ordered pairs of the relation.
- Find the domain of the relation.
- Find the range of the relation.
Domain and Range on Graphs
In 1673, Gottfried Leibniz, the German mathematician who co-invented calculus, seems to be the first person to use the word
function
in a mathematical sense, although his use of it does not exactly fit with the modern use and definition. The person who is credited with the modern definition of function is Swiss mathematician Johann Bernoulli, who wrote about it in a letter to Leibniz in 1698. Supposedly, Leibniz wrote Bernoulli back, approving of this use of the word. In 1734, the use of the notation
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7.9: Ampere’s Law (Magnetostatics) - Differential Form
-
- Last updated
- Save as PDF
The integral form of Amperes’ Circuital Law (ACL) for magnetostatics relates the magnetic field along a closed path to the total current flowing through any surface bounded by that path. In mathematical form:
\[\oint_{\mathcal C}{ {\bf H} \cdot d{\bf l} } = I_{encl} \nonumber \]
where \({\bf H}\) is magnetic field intensity, \({\mathcal C}\) is the closed curve, and \(I_{encl}\) is the total current flowing through any surface bounded by \({\mathcal C}\). In this section, we derive the differential form of this equation. In some applications, this differential equation, combined with boundary conditions associated with discontinuities in structure and materials, can be used to solve for the magnetic field in arbitrarily complicated scenarios. A more direct reason for seeking out this differential equation is that we gain a little more insight into the relationship between current and the magnetic field, disclosed at the end of this section.
The equation we seek may be obtained using Stokes’ Theorem, which in the present case may be written:
\[\int_{\mathcal S} \left( \nabla \times {\bf H} \right) \cdot d{\bf s} = \oint_{\mathcal C} {\bf H}\cdot d{\bf l} \label{m0118_eACL2} \]
where \({\mathcal S}\) is any surface bounded by \({\mathcal C}\), and \(d{\bf s}\) is the differential surface area combined with the unit vector in the direction determined by the right-hand rule from Stokes’ Theorem. ACL tells us that the right side of the above equation is simply \(I_{encl}\). We may express \(I_{encl}\) as the integral of the volume current density \({\bf J}\) (units of A/m\(^2\); Section 6.2) as follows:
\[I_{encl} = \int_{\mathcal S} {\bf J} \cdot d{\bf s} \nonumber \]
so we may rewrite Equation \ref{m0118_eACL2} as follows:
\[\int_{\mathcal S} \left( \nabla \times {\bf H} \right) \cdot d{\bf s} = \int_{\mathcal S} {\bf J} \cdot d{\bf s} \nonumber \]
The above relationship must hold regardless of the specific location or shape of \({\mathcal S}\). The only way this is possible for all possible surfaces in all applicable scenarios is if the integrands are equal. Thus, we obtain the desired expression:
\[\boxed{ \nabla \times {\bf H} = {\bf J} } \label{m0118_eACL} \]
That is, the curl of the magnetic field intensity at a point is equal to the volume current density at that point. Recalling the properties of the curl operator – in particular, that curl involves derivatives with respect to direction – we conclude:
The differential form of Ampere’s Circuital Law for magnetostatics (Equation \ref{m0118_eACL}) indicates that the volume current density at any point in space is proportional to the spatial rate of change of the magnetic field and is perpendicular to the magnetic field at that point.


16.4G: The Iron Cycle
- Compare the terrestrial and marine iron cycles
Iron (Fe) follows a geochemical cycle like many other nutrients. Iron is typically released into the soil or into the ocean through the weathering of rocks or through volcanic eruptions.
The Terrestrial Iron Cycle: In terrestrial ecosystems, plants first absorb iron through their roots from the soil. Iron is required to produce chlorophyl, and plants require sufficient iron to perform photosynthesis. Animals acquire iron when they consume plants, and iron is utilized by vertebrates in hemoglobin, the oxygen-binding protein found in red blood cells. Animals lacking in iron often become anemic and cannot transmit adequate oxygen. Bacteria then release iron back into the soil when they decompose animal tissue.
The Marine Iron Cycle: The oceanic iron cycle is similar to the terrestrial iron cycle, except that the primary producers that absorb iron are typically phytoplankton or cyanobacteria. Iron is then assimilated by consumers when they eat the bacteria or plankton. The role of iron in ocean ecosystems was first discovered when English biologist Joseph Hart noticed “desolate zones,” which are regions that lacked plankton but were rich in nutrients. He hypothesized that iron was the limiting nutrient in these areas. In the past three decades there has been research into using iron fertilization to promote alagal growth in the world’s oceans. Scientists hoped that by adding iron to ocean ecosystems, plants might grown and sequester atmospheric CO2. Iron fertilization was thought to be a possible method for removing the excess CO2 responsible for climate change. Thus far, the results of iron fertilization experiments have been mixed, and there is concern among scientists about the possible consequences of tampering nutrient cycles.
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- Methanogenesis. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Methanogenesis . License : CC BY-SA: Attribution-ShareAlike
- Archea. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Archea . License : CC BY-SA: Attribution-ShareAlike
- syntrophy. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/syntrophy . License : CC BY-SA: Attribution-ShareAlike
- methanogenesis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/methanogenesis . License : CC BY-SA: Attribution-ShareAlike
- File:Carbon cycle-cute diagram.svg - Wikipedia, the free encyclopedia. Provided by : Wikipedia. Located at : en.Wikipedia.org/w/index.php?...ram.svg&page=1 . License : Public Domain: No Known Copyright
- NASA | A Year in the Life of Earth's CO2. Located at : http://www.youtube.com/watch?v=x1SgmFa0r04 . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Carbon cycle. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Carbon_cycle.jpg . License : Public Domain: No Known Copyright
- Coptotermes formosanus shiraki USGov k8204-7. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Co...ov_k8204-7.jpg . License : Public Domain: No Known Copyright
- hypertrophication. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/hypertrophication . License : CC BY-SA: Attribution-ShareAlike
- File:Carbon cycle-cute diagram.svg - Wikipedia, the free encyclopedia. Provided by : Wikipedia. Located at : en.Wikipedia.org/w/index.php?...ram.svg&page=1 . License : Public Domain: No Known Copyright
- NASA | A Year in the Life of Earth's CO2. Located at : http://www.youtube.com/watch?v=x1SgmFa0r04 . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Carbon cycle. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Carbon_cycle.jpg . License : Public Domain: No Known Copyright
- Coptotermes formosanus shiraki USGov k8204-7. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Co...ov_k8204-7.jpg . License : Public Domain: No Known Copyright
- Potomac green water. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Po...reen_water.JPG . License : CC BY-SA: Attribution-ShareAlike
- Phoscycle-EPA. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Phoscycle-EPA.jpg . License : Public Domain: No Known Copyright
- de-nitrification. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/de-nitrification . License : CC BY-SA: Attribution-ShareAlike
- Nitrogen cycle. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Nitrogen_cycle . License : CC BY-SA: Attribution-ShareAlike
- Nitrification. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Nitrification . License : CC BY-SA: Attribution-ShareAlike
- Nitrogen fixation. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Nitrogen_fixation . License : CC BY-SA: Attribution-ShareAlike
- ammonification. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/ammonification . License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//biology/de.../nitrification . License : CC BY-SA: Attribution-ShareAlike
- File:Carbon cycle-cute diagram.svg - Wikipedia, the free encyclopedia. Provided by : Wikipedia. Located at : en.Wikipedia.org/w/index.php?...ram.svg&page=1 . License : Public Domain: No Known Copyright
- NASA | A Year in the Life of Earth's CO2. Located at : http://www.youtube.com/watch?v=x1SgmFa0r04 . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Carbon cycle. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Carbon_cycle.jpg . License : Public Domain: No Known Copyright
- Coptotermes formosanus shiraki USGov k8204-7. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Co...ov_k8204-7.jpg . License : Public Domain: No Known Copyright
- Potomac green water. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Po...reen_water.JPG . License : CC BY-SA: Attribution-ShareAlike
- Phoscycle-EPA. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Phoscycle-EPA.jpg . License : Public Domain: No Known Copyright
- Nitrogen Cycle. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ogen_Cycle.svg . License : CC BY-SA: Attribution-ShareAlike
- Sulfate reducing bacteria. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Sulfate...ucing_bacteria . License : CC BY-SA: Attribution-ShareAlike
- Sulfur cycle. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Sulfur_cycle . License : CC BY-SA: Attribution-ShareAlike
- Sulfur-reducing bacteria. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Sulfur-reducing_bacteria . License : CC BY-SA: Attribution-ShareAlike
- Sulfur metabolism. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Sulfur_metabolism . License : CC BY-SA: Attribution-ShareAlike
- assimilatory sulfate reduction. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/assimil...te%20reduction . License : CC BY-SA: Attribution-ShareAlike
- extremophile. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/extremophile . License : CC BY-SA: Attribution-ShareAlike
- File:Carbon cycle-cute diagram.svg - Wikipedia, the free encyclopedia. Provided by : Wikipedia. Located at : en.Wikipedia.org/w/index.php?...ram.svg&page=1 . License : Public Domain: No Known Copyright
- NASA | A Year in the Life of Earth's CO2. Located at : http://www.youtube.com/watch?v=x1SgmFa0r04 . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Carbon cycle. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Carbon_cycle.jpg . License : Public Domain: No Known Copyright
- Coptotermes formosanus shiraki USGov k8204-7. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Co...ov_k8204-7.jpg . License : Public Domain: No Known Copyright
- Potomac green water. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Po...reen_water.JPG . License : CC BY-SA: Attribution-ShareAlike
- Phoscycle-EPA. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Phoscycle-EPA.jpg . License : Public Domain: No Known Copyright
- Nitrogen Cycle. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ogen_Cycle.svg . License : CC BY-SA: Attribution-ShareAlike
- Ecology/Biogeochemical cycles. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Ecology...Mineral_cycles . License : CC BY-SA: Attribution-ShareAlike
- Iron fertilization. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Iron_fertilization . License : CC BY-SA: Attribution-ShareAlike
- hemogloben. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/hemogloben . License : CC BY-SA: Attribution-ShareAlike
- File:Carbon cycle-cute diagram.svg - Wikipedia, the free encyclopedia. Provided by : Wikipedia. Located at : en.Wikipedia.org/w/index.php?title=File:Carbon_cycle-cute_diagram.svg&page=1 . License : Public Domain: No Known Copyright
- NASA | A Year in the Life of Earth's CO2. Located at : http://www.youtube.com/watch?v=x1SgmFa0r04 . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Carbon cycle. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Carbon_cycle.jpg . License : Public Domain: No Known Copyright
- Coptotermes formosanus shiraki USGov k8204-7. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Coptotermes_formosanus_shiraki_USGov_k8204-7.jpg . License : Public Domain: No Known Copyright
- Potomac green water. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Po...reen_water.JPG . License : CC BY-SA: Attribution-ShareAlike
- Phoscycle-EPA. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Phoscycle-EPA.jpg . License : Public Domain: No Known Copyright
- Nitrogen Cycle. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ogen_Cycle.svg . License : CC BY-SA: Attribution-ShareAlike
- Provided by : Wikimedia. Located at : upload.wikimedia.org/wikipedi...hore_bloom.jpg . License : Public Domain: No Known Copyright
Key Points
- Iron is an important limiting nutrient for plants, which use it to produce chlorophyll. Photosynthesis depends on adequate iron supply. Plants assimilate iron from the soil into their roots.
- Animals consume plants and use the iron to produce hemoglobin, the oxygen transports protein found in red blood cells. When animals die, decomposing bacteria return iron to the soil.
- The marine iron cycle is very similar to the terrestrial iron cycle, except that phytoplankton and cyanobacteria assimilate iron.
- Iron fertilization has been studied as a method for sequestering carbon. Scientists have hoped that by adding iron to the ocean, plankton might be able to sequester the excess CO2 responsible for climate change. However, there is concern about the long term effects of this strategy.
Key Terms
- hemoglobin : the iron-containing oxygen transport metalloprotein in the red blood cells of all vertebrates


11.3: Soil Properties
Soils can be enormously complex systems of organic and inorganic components. Here, we'll concentrate on a few of the most significant properties, texture, structure, color, and chemistry
Soil Texture
Soil texture refers to the relative proportion of sand, silt and clay size particles in a sample of soil. Clay size particles are the smallest being less than .002 mm in size. Silt is a medium size particle falling between .002 and .05 mm in size. The largest particle is sand with diameters between .05 for fine sand to 2.0 mm for very coarse sand. Soils that are dominated by clay are called fine textured soils while those dominated by larger particles are referred to as coarse textured soils. Soil scientists group soil textures into soil texture classes. A soil texture triangle is used to classify the texture class (see below).
The sides of the soil texture triangle are scaled for the percentages of sand, silt, and clay. Clay percentages on the left side of the triangle are read from left to right across the triangle (dashed lines). Silt runs from the top to the bottom along the right side and is read from the upper right to lower left (light, dotted lines). The percentage of sand increases from right to left along the base of the triangle. Sand is read from the lower right towards the upper left portion of the triangle (bold, solid lines). The boundaries of the soil texture classes are highlighted in blue. The intersection of the three sizes on the triangle give the texture class. For instance, if you have a soil with 20% clay, 60% silt, and 20% sand it falls in the "silt loam" class.
What is the texture class for a soil having 45% clay, 45% silt, and 10% sand?
Slit Loam, Clay, Silty Clay, or Silty Clay Loam?
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Silty Clay
Soil texture effects many other properties like structure, chemistry, and most notably, soil porosity, and permeability. Soil porosity refers to the amount of pore, or open space between soil particles. Pores are created by the contacts made between irregular shaped soil particles. Fine textured soil has more pore space than coarse textured because you can pack more small particles into a unit volume than larger ones. More particles in a unit volume creates more contacts between the irregular shaped surfaces and hence more pore space. As a result, fine textured clay soils hold more water than coarse textured sandy soils. Permeability is the degree of connectivity between soil pores. A highly permeable soil is one in which water runs through it quite readily. Coarse textured soils tend to have large, well-connected pore spaces and hence high permeability.
Soil Structure
Soil structure is the way soil particles aggregate together into what are called peds. Peds come in a variety of shapes depending on the texture, composition, and environment.
Granular , or crumb structures, look like cookie crumbs. They tend to form an open structure that allows water and air to penetrate the soil. Platy structure looks like stacks of dinner plates overlaying one another. Platy structure tends to impede the downward movement of water and plant roots through the soil. Therefore, open structures tend to be better agricultural soils.
Bulk density of a soil is the mass per unit volume including the pore space. Bulk density increases with clay content and is considered a measure of the compactness of the soil. The greater the bulk density, the more compact the soil. Compact soils have low permeability, inhibiting the movement of water. The use of heavy agricultural equipment can cause compaction of soil, especially in wet clay soil. Soil compaction results in reduced infiltration and increase runoff and erosion.
Soil Chemistry
As plant material dies and decays it adds organic matter in the form of humus to the soil. Humus improves soil moisture retention while affecting soil chemistry. Cations such as calcium, magnesium, sodium, and potassium are attracted and held to humus. These cations are rather weakly held to the humus and can be replaced by metallic ions like iron and aluminum, releasing them into the soil for plants to use. Soils with the ability to absorb and retain exchangeable cations have a high cation-exchange capacity. Soils with a high cation-exchange capacity are more fertile than those with a low exchange capacity.
Hydrogen ion concentration in the soil is measured in terms of the pH scale . Soil pH ranges from 3 to 10. Pure water has a pH of 7 which is considered neutral, pH values greater than seven are considered basic or alkaline , below seven acidic . Most good agricultural soils have a pH between 5 and 7. Though acidic soils pose a problem for agriculture due to their lack of nutrients, alkaline soils can pose a problem as well. Alkaline soils may contain appreciable amounts of sodium that exceed the tolerances of plants, contribute to high bulk density and poor soil structure. Alkaline soils are common in semiarid regions.


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5: Vector Fields, Line Integrals, and Vector Theorems
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5.1: Introduction to Vector Field Chapter
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Vector fields have many applications because they can be used to model real fields such as electromagnetic or gravitational fields. A deep understanding of physics or engineering is impossible without an understanding of vector fields. Furthermore, vector fields have mathematical properties that are worthy of study in their own right. In particular, vector fields can be used to develop several higher-dimensional versions of the Fundamental Theorem of Calculus.
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5.2: Vector Fields
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Vector fields are an important tool for describing many physical concepts, such as gravitation and electromagnetism, which affect the behavior of objects over a large region of a plane or of space. They are also useful for dealing with large-scale behavior such as atmospheric storms or deep-sea ocean currents. In this section, we examine the basic definitions and graphs of vector fields so we can study them in more detail in the rest of this chapter.
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5.3: Line Integrals
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Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see.Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see.
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5.4: Conservative Vector Fields
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In this section, we continue the study of conservative vector fields. We examine the Fundamental Theorem for Line Integrals, which is a useful generalization of the Fundamental Theorem of Calculus to line integrals of conservative vector fields. We also discover show how to test whether a given vector field is conservative, and determine how to build a potential function for a vector field known to be conservative.
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5.5: Green's Theorem
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Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region D in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve C and a double integral over the region enclosed by C.
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5.6: Divergence and Curl
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Divergence and curl are two important operations on a vector field. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus. In addition, curl and divergence appear in mathematical descriptions of fluid mechanics, electromagnetism, and elasticity theory, which are important concepts in physics and engineering.
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5.7: Surface Integrals
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If we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral. We can extend the concept of a line integral to a surface integral to allow us to perform this integration. Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to expand the Fundamental Theorem of Calculus to higher dimensions.
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5.8: Stokes' Theorem
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In this section, we study Stokes’ theorem, a higher-dimensional generalization of Green’s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green’s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions. Stokes’ theorem relates a vector surface integral over surface S in space to a line integral around the boundary of S.
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5.9: The Divergence Theorem
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We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain. In this section, we state the divergence theorem, which is the final theorem of this type that we will study.


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4.1: Prelude to Discrete Random Variables
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Random Variable (RV) a characteristic of interest in a population being studied
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4.2: Probability Distribution Function (PDF) for a Discrete Random Variable
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A discrete probability distribution function has two characteristics: Each probability is between zero and one, inclusive. The sum of the probabilities is one.
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4.3: Mean or Expected Value and Standard Deviation
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The expected value is often referred to as the "long-term" average or mean. This means that over the long term of doing an experiment over and over, you would expect this average. This “long-term average” is known as the mean or expected value of the experiment and is denoted by the Greek letter μμ . In other words, after conducting many trials of an experiment, you would expect this average value.
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4.4: Binomial Distribution
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A statistical experiment can be classified as a binomial experiment if the following conditions are met: (1) There are a fixed number of trials. (2)There are only two possible outcomes: "success" or "failure" for each trial. (3) The trials are independent and are repeated using identical conditions. The outcomes of a binomial experiment fit a binomial probability distribution.
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4.5: Geometric Distribution
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There are three characteristics of a geometric experiment: (1) There are one or more Bernoulli trials with all failures except the last one, which is a success. (2) In theory, the number of trials could go on forever. There must be at least one trial. (3) The probability, p, of a success and the probability, q, of a failure are the same for each trial. In a geometric experiment, define the discrete random variable X as the number of independent trials until the first success.
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4.6: Hypergeometric Distribution
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A hypergeometric experiment is a statistical experiment with the following properties: You take samples from two groups. You are concerned with a group of interest, called the first group. You sample without replacement from the combined groups. Each pick is not independent, since sampling is without replacement. You are not dealing with Bernoulli Trials. The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution.
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4.7: Poisson Distribution
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A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is "small" (less than or equal to 0.05) and the number of trials is "large" (greater than or equal to 20).
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4.8: Discrete Distribution (Playing Card Experiment)
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A statistics Worksheet: The student will compare empirical data and a theoretical distribution to determine if an everyday experiment fits a discrete distribution. The student will demonstrate an understanding of long-term probabilities.
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4.9: Discrete Distribution (Lucky Dice Experiment)
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A statistics Worksheet: The student will compare empirical data and a theoretical distribution to determine if a Tet gambling game fits a discrete distribution. The student will demonstrate an understanding of long-term probabilities.
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4.E: Discrete Random Variables (Exercises)
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These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.


7.10: Rings, Moons, and Pluto (Exercises)
For Further Exploration
Articles
Moons
Carroll, M. “Titan: What We’ve Learned about a Strange New World.” Astronomy (March 2010): 30. Nice review of Cassini mission results.
Elliot, J. “The Warming Wisps of Triton.” Sky & Telescope (February 1999): 42. About Neptune’s intriguing moon.
Hayes, A., “Secrets from Titan’s Seas.” Astronomy (October 2015): 24. Good review of what we now know and what puzzles us about the hydrocarbon lakes of Titan.
Jewitt, D., et al. “The Strangest Satellites in the Solar System.” Scientific American (August 2006): 40. Small irregular moons in the outer solar system.
Lakdawalla, E. “Ice Worlds of the Ringed Planet.” Sky & Telescope (June 2009): 27. On the Cassini mission exploration of Enceladus, Iapetus, and other moons.
Mackenzie, D. “Is There Life under the Ice?” Astronomy (August 2001): 32. On future exploration of Europa.
Robertson, D. “Where Goes the Rain?” Sky & Telescope (March 2013): 26. About the methane weather cycle on Titan and what Cassini experiments are telling us.
Scharf, C. “A Universe of Dark Oceans.” Sky & Telescope (December 2014): 20. Subsurface oceans on Europa, Ganymede, Enceladus, and Titan.
Showalter, M. “How to Catch a Moon (or Two) of Pluto.” Astronomy Beat (December 2012): www.astrosociety.org/wp-conte...ab2012-106.pdf. On the discovery of small moons around Pluto, written by the person who discovered two of them.
Spencer, J. “Galileo’s Closest Look at Io.” Sky & Telescope (May 2001): 40.
Talcott, R. “Cassini Flies through Enceladus’ Geysers.” Astronomy (March 2009): 32.
Zimmerman, R. “Does Methane Flow on Titan?” Astronomy (February 2014): 22. Ideas about lakes, channels, and rain.
Pluto
Stern, A. “Pluto: Up Close and Personal.” Astronomy (July 2015): 22. Good summary of the history of understanding Pluto and our current knowledge on the eve of the New Horizons encounter.
Stern, A. “The Pluto System Explored.” Astronomy (November 2015): 24. Fine review of what the team learned from the first few data downloads from New Horizons.
Tombaugh, C. “How I Found Pluto” Astronomy Beat (May 2009): astrosociety.org/wp-content/u.../ab2009-23.pdf.
Rings
Beatty, J. “Saturn’s Amazing Rings.” Sky & Telescope (May 2013): 18. Good 7-page summary of what we know.
Burns, J., et al. “Bejeweled Worlds.” Scientific American (February 2002): 64. On rings throughout the solar system.
Elliot, J., et al. “Discovering the Rings of Uranus.” Sky & Telescope (June 1977): 412.
Esposito, L. “The Changing Shape of Planetary Rings.” Astronomy (September 1987): 6.
Sobel, D. “Secrets of the Rings.” Discover (April 1994): 86. Discusses the outer planet ring systems.
Tiscareno, M. “Ringworld Revelations.” Sky & Telescope (February 2007): 32. Cassini results about the rings of Saturn.
Websites
Note: Many of the sites about planets and planetary missions listed for Other Worlds: An Introduction to the Solar System and The Giant Planets also include good information about the moons of the planets.
Cassini Mission to Saturn: http://saturn.jpl.nasa.gov/ and www.esa.int/SPECIALS/Cassini-...ens/index.html and ciclops.org
Jupiter’s Moons, at JPL: http://solarsystem.nasa.gov/planets/jupiter/moons
Neptune’s Moons, at JPL: http://solarsystem.nasa.gov/planets/neptune/moons
New Horizons Mission: http://pluto.jhuapl.edu . Gives the latest news bulletins and images from the Pluto encounter, plus lots of background information.
Pluto, at JPL: http://solarsystem.nasa.gov/planets/pluto
Saturn’s Moons, at JPL: http://solarsystem.nasa.gov/planets/saturn/moons
Uranus’ Moons, at JPL: http://solarsystem.nasa.gov/planets/uranus/moons
Apps
Two apps you can buy for iPhones or iPads can show you the positions and features of the moons of Jupiter and Saturn for any selected date:
- Jupiter Atlas: https://itunes.apple.com/us/app/ju[i...352033947?mt=8
- Saturn Atlas: https://itunes.apple.com/us/app/satu...352038051?mt=8
Videos
Amazing Moons: https://www.youtube.com/watch?v=CQjZf2bW9XQ . 2016 NASA video on intriguing moons in our solar system (4:16).
Briny Breath of Enceladus: http://www.jpl.nasa.gov/video/details.php?id=846 . Brief 2009 JPL film on the geysers of Enceladus (2:36).
Dr. Carolyn Porco’s TED Talk on Enceladus: https://www.youtube.com/watch?v=TRQdHrGuVgI (3:26).
Titan: http://www.youtube.com/watch?v=iTrOFefYxFg . Video from Open University, with interviews, animations, and images (8:11).
Europa Mission: http://www.jpl.nasa.gov/events/lectu...r=2016&month=2 . 2016 talk by two JPL scientists on NASA’s plans for a mission to Jupiter’s moon, which may have an underground liquid ocean (1:26:22).
Great Planet Debate: http://gpd.jhuapl.edu/debate/debateStream.php . Neil deGrasse Tyson debates Mark Sykes about how to characterize Pluto, in 2008 (1:14:11).
How I Killed Pluto and Why It Had It Coming: http://www.youtube.com/watch?v=7pbj_llmiMg . 2011 Silicon Valley Astronomy Lecture by Michael Brown on the “demotion” of Pluto to a dwarf planet (1:27:13).
Seeking Pluto’s Frigid Heart: https://www.youtube.com/watch?v=jIxQXGTl_mo . Dramatic 2016 New York Times production, narrated by Dennis Overbye (7:43).
Saturn’s Restless Rings: https://www.youtube.com/watch?v=X5zcrEze8L4 . 2013 talk by Mark Showalter in the Silicon Valley Astronomy Lecture Series (1:30:59).
Review Questions
- What are the moons of the outer planets made of, and how is their composition different from that of our Moon?
- Compare the geology of Callisto, Ganymede, and Titan.
- What is the evidence for a liquid water ocean on Europa, and why is this interesting to scientists searching for extraterrestrial life?
- Explain the energy source that powers the volcanoes of Io.
- Compare the properties of Titan’s atmosphere with those of Earth’s atmosphere.
- How was Pluto discovered? Why did it take so long to find it?
- How are Triton and Pluto similar?
- Describe and compare the rings of Saturn and Uranus, including their possible origins.
- Why were the rings of Uranus not observed directly from telescopes on the ground on Earth? How were they discovered?
- List at least three major differences between Pluto and the terrestrial planets.
- The Hubble Space Telescope images of Pluto in 2002 showed a bright spot and some darker areas around it. Now that we have the close-up New Horizons images, what did the large bright region on Pluto turn out to be?
- Saturn’s E ring is broad and thin, and far from Saturn. It requires fresh particles to sustain itself. What is the source of new E-ring particles?
Thought Questions
- Why do you think the outer planets have such extensive systems of rings and moons, while the inner planets do not?
- Ganymede and Callisto were the first icy objects to be studied from a geological point of view. Summarize the main differences between their geology and that of the rocky terrestrial planets.
- Compare the properties of the volcanoes on Io with those of terrestrial volcanoes. Give at least two similarities and two differences.
- Would you expect to find more impact craters on Io or Callisto? Why?
- Why is it unlikely that humans will be traveling to Io?
- Why do you suppose the rings of Saturn are made of bright particles, whereas the particles in the rings of Uranus and Neptune are black?
- Suppose you miraculously removed all of Saturn’s moons. What would happen to its rings?
- We have a lot of good images of the large moons of Jupiter and Saturn from the Galileo and Cassini spacecraft missions (check out NASA’s Planetary Photojournal site, at http://photojournal.jpl.nasa.gov , to see the variety). Now that the New Horizons mission has gone to Pluto, why don’t we have as many good images of all sides of Pluto and Charon?
- In the Star Wars movie Star Wars Episode VI: Return of the Jedi , a key battle takes place on the inhabited “forest moon” Endor, which supposedly orbits around a gas giant planet. From what you have learned about planets and moons of the solar system, why would this be an unusual situation?